1 Answer
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It is difficult to perceive where you get stuck:
$$\begin{align*}p(b|x)&\propto \dfrac{b^a}{\Gamma (a)}x^{a-1}e^{-xb}\cdot\dfrac{\sqrt{a}}{b} \\ &\propto b^{a-1} e^{-xb} \\ &\propto \dfrac{x^a\,b^{a-1}}{\Gamma(a)}\,e^{-xb}\end{align*}$$
which shows the posterior is a Gamma $\mathcal{G}(a,x)$ distribution.

$\begingroup$I didn't see that $\sqrt{a}$ could be absorbed into the proportionality. Later it seems that $x^a$ appears from nowhere... Is this allowed because in the posterior we are conditioning on $x$, which means that it is fixed (constant)?$\endgroup$
– ADBFeb 18 '16 at 7:45

$\begingroup$Indeed, everything but $b$ is a constant for the proportionality $\propto$ sign.$\endgroup$
– Xi'anFeb 18 '16 at 8:39